Final-state rescattering mechanism of the $\Delta(1232)^{++}$ production in $\Lambda^+_c \to K^- \pi^+ p$ decay

Abstract

We investigate the production of the $\Delta(1232)^{++}$ resonance in the charmed baryon weak decay $\Lambda^+_c \to K^- \pi^+ p$, focusing on the $\pi^+ p$ final-state rescattering mechanism. The direct $W^+$ exchange diagram is expected to be suppressed, hence we adopt the $W^+$ internal emission process $\Lambda^+_c \to p \bar K^{*0}(892)$ followed by the subsequent decay $\bar{K}^{*0} \to K^- \pi^+$ as the dominant source of the final state particles. The $\Delta(1232)^{++}$ resonance is then generated via $\pi^+ p$ rescattering within a triangle loop mechanism. Our calculations incorporate both the tree-level $\bar K^{*0}(892)$ and the dynamically generated $\bar{K}^*_0(700)$ state arising from the $S$-wave $K \pi$ final state interaction. We find that our theoretical results can reproduce the bump and peak structures in the $K^- \pi^+$ invariant mass distributions for the $\bar{K}^*_0(700)$ and $\bar{K}^{*0}(892)$, respectively. Meanwhile, the peak for the $\Delta(1232)^{++}$ in the $\pi^+ p$ invariant mass distributions is also well described. The $\Delta(1232)^{++}$ signal naturally emerges from rescattering effects, and adopting the pole parameters of $\Delta(1232)$ resonance yields an improved description of the experimental data. In addition, we obtain a branching fraction ratio $\mathcal{B}[\Lambda_c^+ \to \Delta(1232)^{++} K^-] / \mathcal{B}[\Lambda_c^+ \to p \bar{K}^{*0}(892)] \approx 0.5$, which is lower than the experimentally measured value. This discrepancy suggests that interference effects are likely significant in this decay process. Future high-precision measurements will further verify the proposed rescattering mechanism.